An improved upper bound on the independent double Roman domination number of trees

Abstract

For a graph an independent double Roman dominating function (IDRDF) is a function having the property that: (i) every vertex with f(v) = 0 has a neighbor u with f(u) = 3 or at least two neighbors x and y such that (ii) every vertex with f(v) = 1 has at least one neighbor assigned a 2 or 3 under f; (iii) the set of vertices assigned non-zero values under f is independent. The weight of an IDRDF is the sum of its values overs all vertices, and the independent double Roman domination number is the minimum weight of an IDRDF on In this article, we show that for every tree T of order where s(T) is the number of support vertices of T, improving the -upper bound established in [Maimani et al. Independent double Roman domination in graphs, Bulletin of the Iranian Mathematical Society, 46 (2020) 543–555]. Moreover, we characterize the trees T of order with